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The formation of large social groups having uniform opinions influenced by mass media is currently an important topic in the social sciences. In this work, we explore and extend an off-lattice, two-dimensional Potts model (Eur. Phys. J. B 87, 78 [2014]) that describes the formation and dynamics of opinions in social groups according to individual consequence and agreement between neighbors. This model was originally obtained by the application of the maximum entropy principle, a general method in statistical inference, and using the same methodology we have now included the influence of mass media as a constant external field. By means of microcanonical Monte Carlo Metropolis simulations on a setup with two regions with opposing external influences, we have shown the presence of metastable states associated to the formation of clusters aligned with the locally imposed opinion. Our results suggest that, for some values of the total energy of the system, only a single cluster with a uniform opinion survives, thus the presence of two large, opposing groups is not a thermodynamically stable configuration.

The generalized equipartition theorem known as the conjugate variables theorem (Phys. Rev. E 86, 051136 [2012]), originally obtained in the context of statistical inference of continuous random variables, is extended in this work to the case of discrete variables. Using this new set of theorems we derive novel thermodynamic identities for the canonical ensemble connecting temperature with measurable observables.

Superstatistics is an elegant framework for the description of steady-state thermodynamics, mostly used for systems with long-range interactions such as plasmas. In this work, we show that the potential energy distribution of a classical system under externally imposed energy fluctuations can also be described by superstatistics in the thermodynamic limit. As an example, we apply this formalism to the thermodynamics of a finite Lennard-Jones crystal with constant microcanonical heat capacity driven by sinusoidal energy oscillations. Our results show that molecular dynamics simulations of the Lennard-Jones crystal are in agreement with the provided theoretical predictions.

In this article we derive a useful expectation identity using the language of quantum statistical mechanics, where density matrices represent the state of knowledge about the system. This identity allows to establish relations between different quantum observables depending on a continuous parameter. Such a parameter can be contained in the observables itself (e.g. perturbative parameter) or may appear as a Lagrange multiplier (inverse temperature, chemical potential, etc.) in the density matrix, excluding parameters that modify the underlying Hilbert space. In this way, using both canonical and grand canonical density matrices along with certain quantum observables (Hamiltonian, number operator, the density matrix itself, etc.) we found new identities in the field, showing not only its derivation but also their meaning. Additionally, we found that some theorems of traditional quantum statistics and quantum chemistry, such as the thermodynamical fluctuation-dissipation theorem, the Ehrenfest, and the Hellmann-Feynman theorems, among others, are particular instances of our aforementioned quantum expectation identity. At last, using a generalized density matrix arising from the Maximum-Entropy principle, we derive generalized quantum expectation identities: these generalized identities allow us to group all the previous cases in a unitary scheme., Comment: 27 pages, 3 tables, 3 figures

Non-equilibrium systems in steady states are commonly described by generalized statistical mechanical theories such as non-extensive statistics and superstatistics. Superstatistics assumes that the inverse temperature $\beta = 1/(k_B T)$ follows some pre-established statistical distribution, however, it has been previously proved (Physica A 505, 864-870 [2018]) that $\beta$ cannot be associated to an observable function $B(\boldsymbol{\Gamma})$ of the microstates $\boldsymbol{\Gamma}$. In this work, we provide an information-theoretical interpretation of this theorem by introducing a new quantity $\mathcal{D}$, the mutual information between $\beta$ and $\boldsymbol{\Gamma}$. Our results show that $\mathcal{D}$ is also a measure of departure from canonical equilibrium, and reveal a minimum, non-zero uncertainty about $\beta$ given $\boldsymbol{\Gamma}$ for every non-canonical superstatistical ensemble. This supports the use of the mutual information as a descriptor of complexity and correlation in complex systems, also providing in some cases a sound basis for the use of Tsallis' entropic index $q$ as a measure of distance from equilibrium, being in those cases a proxy for $\mathcal{D}$.

In recent years, the theory of superstatistics, which aims to describe non-equilibrium steady state systems, has gained attention due to its different real world applications, highlighting its versatility and concise mathematical formulation in terms of a probability density for the inverse temperature $\beta=1/k_{B}T$. When exploring the domain of application of the superstatistical theory, recent works have shown some necessary conditions for a superstatistical description of a given steady state, in terms of the fundamental and microcanonical inverse temperature. In this work, a new theorem that establishes a sufficient condition for the existence of a superstatistical description of a particular steady state is presented, using the language of moment-generating functions and connecting them with properties of the derivatives of the fundamental inverse temperature.

Kappa-distributed velocities in plasmas are common in a wide variety of settings, from low-density to high-density plasmas, and appear in both space and laboratory plasmas. Despite being widely used as a model, the origin of the kappa distribution remains, to this day, unclear. For instance, deviations from the Maxwell-Boltzmann distribution are sometimes regarded as a signature of the non-additivity of the thermodynamic entropy, although there are alternative frameworks such as superstatistics where such an assumption is not needed. In this work we recover the kappa distribution for particle velocities in a non-equilibrium, steady-state plasma from the theory of superstatistics, assuming only a single requirement on the dependence between the kinetic energy of a test particle and that of its immediate environment. Most importantly, unlike previous derivations based on superstatistics, we do not make any assumption about temperature or its statistical distribution, instead obtaining them from the requirement on kinetic energies. Our results suggest that a description of kappa-distributed plasma is simpler in terms of features of the inverse temperature distribution rather than the traditional parameters $\kappa$ and the thermal velocity $v_{\text{th}}$.

A statistical, path-dependent framework to describe time-dependent macroscopic theories using the Principle of Maximum Caliber is presented. By means of this procedure, it is possible to infer predictive non-equilibrium statistical mechanical models from a variational principle, provided that the adequate time-dependent constraints and the state of the system at some specific times are given. The approach is exemplified by obtaining the description of a time-dependent Brownian particle from kinetic restrictions. We relate the predictive nature of a model to the structure of the prior distribution that represents the state of knowledge about the system before the dynamical constraints are considered. Non-predictive models are shown to be possible in the presented framework and as an example, retrodictive dynamics are obtained from the same kinetic constraints.

We analyze the behavior of the microcanonical and canonical caloric curves for a piecewise model of the configurational density of states of simple solids, in the context of melting from the superheated state, as realized numerically in the Z-method via atomistic molecular dynamics. A first-order phase transition with metastable regions is reproduced by the model, being therefore useful to describe aspects of the melting transition. Within this model, transcendental equations connecting the superheating limit, the melting point, and the specific heat of each phase are presented and numerically solved. Our results suggest that the essential elements of the microcanonical Z curves can be extracted from simple modeling of the configurational density of states.

Although generalized ensembles have now been in use in statistical mechanics for decades, including frameworks such as Tsallis' nonextensive statistics and superstatistics, a classification of these generalized ensembles outlining the boundaries of validity of different families of models, is still lacking. In this work, such a classification is proposed in terms of supercanonical and subcanonical ensembles, according to a newly defined parameter, the inverse temperature covariance parameter $\mathcal{U}$. This parameter is non-negative in superstatistics (and is equal to the variance of the inverse temperature) but can be negative for other families of statistical ensembles, adquiring then a broader meaning. It is shown that $\mathcal{U}$ is equal for every region of a composite system in a steady state, and examples are given of supercanonical and subcanonical states.

The theory of superstatistics, originally proposed for the study of complex nonequilibrium systems, has recently been extended to studies of small systems interacting with a finite environment, because such systems display interestingly similar statistical behavior. In both situations there are several applicable definitions of inverse temperature, either intrinsic or dependent of the statistical ensemble. In this work we develop these concepts focusing our attention on a region of an isolated, one-dimensional Ising chain as an example of a subsystem that does not follow the canonical Gibbs distribution. For this example, we explicitly show that superstatistics cannot describe the behavior of the subsystem, and verify a recently reported relation between the fundamental and microcanonical inverse temperatures. Our results hint at a new framework for dealing with regions of microcanonical systems with positive heat capacity, which should be described by some new class of statistical ensembles outside superstatistics but still preserving the notion of temperature fluctuations.

In this work we develop and implement a novel Bayesian method for computing the DOS of a system. This method is based on the use of a test function with adjustable parameters and we use Bayes theorem to find the best parameters given a certain number of measurements done on the system. This measurements can be done in any ensemble defined by a distribution function. We found that the algorithm can find the DOS in a reasonable amount of time, and that if the test function is suitable enough, the DOS found by the algorithm is very close to the true DOS.

Superstatistics is a generalization of equilibrium statistical mechanics that describes systems in nonequilibrium steady states. Among the possible superstatistical distributions, the $q$-canonical ensemble (also known as Tsallis' statistics, and in plasma physics as Kappa distributions) is probably the most widely used, however the current explanations of its origin are not completely consistent. In this work it is shown that, under a Bayesian interpretation of superstatistics, the origin of the $q$-canonical ensemble can be explained as the superstatistical distribution with maximum Shannon-Jaynes entropy under noninformative constraints. The $q$-canonical distributions are singled out by the mathematical structure of superstatistics itself, and thus no assumptions about the physics of the systems of interest, or regarding their complexity or range of interactions, are needed. These results support the thesis that the success of the $q$-canonical ensemble is information-theoretical in nature, and explainable in terms of the original maximum entropy principle by Jaynes.

The existence of fluctuations of temperature has been a somewhat controversial topic in thermodynamics but nowadays it is recognized that they must be taken into account in small, finite systems. Although for nonequilibrium steady states superstatistics is becoming the \textit{de facto} framework for expressing such temperature fluctuations, some recent results put into question the idea of temperature as a phase space observable. In this work we present and explore the statistics that describes a part of an isolated system, small enough to have well-defined uncertainties in energy and temperature, but lacking a superstatistical description. These results motivate the use of the so-called fundamental temperature as an observable and may be relevant for the statistical description of small systems in physical chemistry.

In this work we develop an implementation of the Wang--Landau algorithm [Phys. Rev. Lett. \textbf{86}, 2050-2053 (2001)]. This algorithm allows us to find the density of states (DOS), a function that, for a given system, describes the proportion of states that have a certain energy. The implementation uses the Python language for the algorithm itself, and it can take advantage of any library, such as the powerful LAMMPS library, for the computation of energy. Therefore, the resulting implementation is simple and flexible without sacrificing efficiency. This implementation also considers recent developments in the parallelization of the code for faster computation. We establish the soundness and effectiveness of our implementation by studying well-known systems such as the Ising model, the Lennard--Jones and EAM solids. We have found that our implementation can find the DOS with very good precision in a reasonable amount of time. Therefore, we are equipped with a very powerful and flexible implementation that can be easily used in order to study more realistic models of matter.

The interactions between a group of components are commonly studied in several areas of science (social science, biology, material science, complex dynamical systems, among others) using the methods of thermodynamics and statistical mechanics. In this work we study the properties of the recently proposed off-lattice, two-dimensional Potts model [Eur. Phys. J. B 87, 78 (2014)], originally motivated by the dynamics of agent opinions, and which is described by a Hamiltonian obtained by a maximum entropy inference procedure. We performed microcanonical and canonical Monte Carlo simulations of the first-order phase transition in the model, revealing a caloric curve with metastable regions. Furthermore, we report a "switching" behavior between multiple metastable states. We also note that the thermodynamics of the model has striking similarities with systems having long-range interactions, even though the interactions are short-ranged.

High-energy incident particles from both pulsed and continuous radiation sources can induce significant damage to the structure of a material by creating vacancy-interstitial pairs and other more complex defects, and this leads typically to localized melting. In this work, we present evidence via molecular dynamics simulations of a superheated solid state in BCC tungsten induced by single PKA events of $\sim$ 1.5 keV of energy. Despite the striking difference between this highly inhomogeneous energy injection and homogeneous melting, the lifetime of the obtained superheated solid state, reaching up to 200 ps, is described through a waiting time distribution in agreement with previous studies on superheating in the Z-method methodology.

Superstatistics describes nonequilibrium steady states as superpositions of canonical ensembles with a probability distribution of temperatures. Rather than assume a certain distribution of temperature, recently [J. Phys. A: Math. Theor. 53, 045004 (2020)] we have discussed general conditions under which a system in contact with a finite environment can be described by superstatistics together with a physically interpretable, microscopic definition of temperature. In this work, we present a new interpretation of this result in terms of the standard maximum entropy principle (MaxEnt) using conditional expectation constraints, and provide an example model where this framework can be tested.

The thermodynamic properties of systems with long-range interactions is still an ongoing challenge, both from the point of view of theory as well as computer simulation. In this work we study a model system, a Coulomb gas confined inside a sphere, by using the Wang-Landau algorithm. We have computed the configurational density of states (CDOS), the thermodynamic entropy and the caloric curve, and compared with microcanonical Metropolis simulations, while showing how concepts such as the configurational inverse temperature can be used to understand some aspects of thermodynamic behavior. A dynamical multistability behavior is seen at low energies in microcanonical Monte Carlo simulations, suggesting that flat-histogram methods are in fact superior alternatives to traditional simulation in complex systems.

A permanent challenge in physics and other disciplines is to solve partial differential equations, thereby a beneficial investigation is to continue searching for new procedures to do it. In this Letter, a novel Monte-Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the paths space with a probability functional obtained by using the maximum caliber principle. The methodology was applied to the free particle and the harmonic oscillator problems, where the numerically-averaged path obtained from the Monte-Carlo simulation converges to the analytical solution from classical mechanics, in an analogous way with a canonical system where energy is minimized by sampling the state space and computing the average state for each system. Thus, we expect that this procedure can be general enough to solve other differential equations in physics and to be a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.

The distribution of waiting times until the occurrence of a critical event is a crucial statistical problem across several disciplines in Science. In this work we present a statistical model in which a relevant quantity X accumulates until overcoming a threshold X*, which defines the collapse. The obtained waiting time distribution is a mixture of gamma distributions, which in turn can be approximated as an effective gamma distribution.

Nonextensive statistics is a formalism of statistical mechanics that describes the ocurrence of power-law distributions in complex systems, particularly the so-called $q$ exponential family of distributions. In this work we present the use of fluctuation theorems for $q$-canonical ensembles as a powerful tool to readily obtain statistical properties. In particular, we have obtained strong conditions for the possible values of $q$ depending on the density of states of the system.

Superstatistics is a framework in nonequilibrium statistical mechanics that successfully describes a wide variety of complex systems, including hydrodynamic turbulence, weakly-collisional plasmas, cosmic rays, power grid fluctuations, among several others. In this work we analyze the class of nonequilibrium steady-state systems consisting of a subsystem and its environment, and where the subsystem is described by the superstatistical framework. In this case we provide an answer to the mechanism by which a broad distribution of temperature arises, namely due to correlation between subsystem and environment. We prove that there is a unique microscopic definition $\mathcal{B}$ of inverse temperature compatible with superstatistics, in the sense that all moments of $\mathcal{B}$ and $\beta=1/(k_B T)$ coincide. The function $\mathcal{B}$ however, cannot depend on the degrees of freedom of the system itself, only on the environment, in full agreement with our previous impossibility theorem [Physica A \textbf{505}, 864-870 (2018)]. The present results also constrain the possible joint ensembles of system and environment compatible with superstatistics.

The correct modelling of velocity distribution functions for particles in steady-state plasmas is a central element in the study of nuclear fusion and also in the description of space plasmas. In this work, a statistical mechanical formalism for the description of collisionless plasmas in a steady state is presented, based solely on the application of the rules of probability and not relying on the concept of entropy. Beck and Cohen's superstatistical framework is recovered as a limiting case, and a "microscopic" definition of inverse temperature $\beta$ is given. Non-extensivity is not invoked a priori but enters the picture only through the analysis of correlations between parts of the system.

Metastable states in first-order phase transitions reveal interesting behavior about a wide range of systems in statistical mechanics, including spin systems, cellular automata and condensed matter systems. These metastable states are often observed in a microcanonical setting, where they manifest long-range correlations due to collective effects. In this work we show the existence of long-range potential energy correlations between atoms in a microcanonical superheated Lennard-Jones crystal prior to homogeneous melting. Our results suggest that the cooperative motion made possible by the presence of vacancy-interstitial pairs above the melting temperature induces effective long-range interatomic forces even beyond the fourth neighboring layer.

The kinematics and dynamics of deterministic physical systems have been a foundation of our understanding of the world since Galileo and Newton. For real systems, however, uncertainty is largely present via external forces such as friction or lack of precise knowledge about the initial conditions of the system. In this work we focus in the latter case and describe the use of inference methodologies in solving the statistical properties of classical systems subject to uncertain initial conditions. In particular we describe the application of the formalism of Maximum Entropy (MaxEnt) inference to the problem of projectile motion given information about the average horizontal range over many realizations. By using MaxEnt we can invert the problem and use the provided information on the average range to reduce the original uncertainty in the initial conditions, while also achieving additional insights based on the shape of the posterior probabilities for the initial conditions probabilities and the projectile path distribution itself. The wide applicability of this procedure, as well as its ease of use, reveals a useful tool by which to revisit a large number of physics problems, from classrooms to frontier research.

Homogeneous melting of superheated crystals at constant energy is a dynamical process, believed to be triggered by the accumulation of thermal vacancies and their self-diffusion. From microcanonical simulations we know that if an ideal crystal is prepared at a given kinetic energy, it takes a random time $t_w$ until the melting mechanism is actually triggered. In this work we have studied in detail the statistics of $t_w$ for melting at different energies by performing a large number of Z-method simulations and applying state-of-the-art methods of Bayesian statistical inference. By focusing on the short-time tail of the distribution function, we show that $t_w$ is actually gamma-distributed rather than exponential (as asserted in previous work), with decreasing probability near $t_w \sim 0$. We also explicitly incorporate in our model the unavoidable truncation of the distribution function due to the limited total time span of a Z method simulation. The probabilistic model presented in this work can provide some insight into the dynamical nature of the homogeneous melting process, as well as giving a well-defined practical procedure to incorporate melting times from simulation into the Z-method in order to reduce the uncertainty in the melting temperature.

We report microcanonical Monte Carlo simulations of melting and superheating of a generic, Lennard-Jones system starting from the crystalline phase. The isochoric curve, the melting temperature $T_m$ and the critical superheating temperature $T_{LS}$ obtained are in close agreement (well within the microcanonical temperature fluctuations) with standard molecular dynamics one-phase and two-phase methods. These results validate the use of microcanonical Monte Carlo to compute melting points, a method which has the advantage of only requiring the configurational degrees of freedom. Our findings show that the strict preservation of the Hamiltonian dynamics does not constitute a necessary condition to produce a realistic estimate of $T_{LS}$ and the melting point, which brings new insight on the nature of the melting transition. These results widen the use and applicability of the recently developed Z method for the determination of the melting points of materials.

For non-equilibrium systems in a steady state we present two necessary and sufficient conditions for the emergence of $q$-canonical ensembles, also known as Tsallis statistics. These conditions are invariance requirements over the definition of subsystem and environment, and over the joint rescaling of temperature and energy. Our approach is independent of, but complementary to, the notions of Tsallis non-extensive statistics and Superstatistics.

The formalism of Quantum Mechanics is derived from the application of Bayesian probability theory to "fragile" systems, i.e. systems that are perturbed by the act of measurement. Complex Hilbert spaces, non-commuting operators and the trace rule for expectations all arise naturally from the use of linear algebra to solve integral equations involving classical probabilities over hidden variables. We comment on the case of non-local hidden variables, where violations of Bell's theorem can be produced, as well as the non-fragile limit of the theory, where all measurements are commutative and the theory becomes analogous to classical statistical mechanics., Comment: 4 pages

The probability axioms by R. T. Cox can be regarded as the modern foundations of Bayesian inference, the idea of assigning degrees of belief to logical propositions in a manner consistent with Boolean logic. In this work it is shown that one can start from an alternative point of view, postulating the existence of an operation for \textit{estimating magnitudes given a state of knowledge}. The properties of this operation can be established through basic consistency requirements. Application of this estimation operation to the truth values of propositions in an integer representation directly yields non-negative, normalized quantities following the sum and product rules of probability.

In the global framework of finding an axiomatic derivation of nonequilibrium Statistical Mechanics from fundamental principles, such as the maximum path entropy -- also known as Maximum Caliber principle -- , this work proposes an alternative derivation of the well-known Jarzynski equality, a nonequilibrium identity of great importance today due to its applications to irreversible processes: biological systems (protein folding), mechanical systems, among others. This equality relates the free energy differences between two equilibrium thermodynamic states with the work performed when going between those states, through an average over a path ensemble. In this work the analysis of Jarzynski's equality will be performed using the formalism of inference over path space. This derivation highlights the wide generality of Jarzynski's original result, which could even be used in non-thermodynamical settings such as social systems, financial and ecological systems., Comment: Conference 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering

In this work we present the fundamental ideas of inference over paths, and show how this formalism implies the continuity equation, which is central for the derivation of the main partial differential equations that constitute non-equilibrium statistical mechanics. Equations such as the Liouville equation, Fokker-Planck equation, among others can be recovered as particular cases of the continuity equation, under different probability fluxes. We derive the continuity equation in its most general form through what we call the \bf time-slicing equation, which lays down the procedure to go from the representation in terms of a path probability functional $\rho[X()]$ to a time-dependent probability density $\rho(x; t)$. The original probability functional $\rho[X()]$ can in principle be constructed from different methods of inference; in this work we sketch an application using the maximum path entropy or maximum caliber principle.

One of the cornerstones in non--equilibrium statistical mechanics (NESM) is Liouville's theorem, a differential equation for the phase space probability $\rho(q,p; t)$. This is usually derived considering the flow in or out of a given surface for a physical system (composed of atoms), via more or less heuristic arguments. In this work, we derive the Liouville equation as the partial differential equation governing the dynamics of the time-dependent probability $\rho(q, p; t)$ of finding a "particle" with Lagrangian $L(q, \dot{q}; t)$ in a specific point $(q, p)$ in phase space at time $t$, with $p=\partial L/\partial \dot{q}$. This derivation depends only on considerations of inference over a space of continuous paths. Because of its generality, our result is valid not only for "physical" systems but for any model depending on constrained information about position and velocity, such as time series.

Given a probability density $P({\bf x}|{\boldsymbol \lambda})$, where $\bf x$ represents continuous degrees of freedom and $\lambda$ a set of parameters, it is possible to construct a general identity relating expectations of observable quantities, which is a generalization of the equipartition theorem in Thermodynamics. In this work we explore some of the consequences of this relation, both in the context of sampling distributions and Bayesian posteriors, and how it can be used to extract some information without the need for explicit calculation of the partition function (or the Bayesian evidence, in the case of posterior expectations). Together with the general family of fluctuation theorems it constitutes a powerful tool for Bayesian/MaxEnt problems.

Superstatistics (Physica A 322, 267-275, 2003) is a formalism that attempts to explain the presence of distributions other than the Boltzmann-Gibbs distributions in Nature, typically power-law behavior, for systems out of equilibrium such as fluids under turbulence, plasmas and gravitational systems. Superstatistics postulates that those systems are found in a superposition of canonical ensembles at different temperatures. The usual interpretation is one of local thermal equilibrium (LTE) in the sense of an inhomogeneous temperature distribution in different regions of space or instants of time. Here we show that, in order for superstatistics to be internally consistent, it is impossible to define a phase-space function or observable $B(p, q)$ corresponding one-to-one to the local value of $\beta=1/k_B T$. Temperature then belongs to a different class of observables than the energy, which has as a phase-space function the Hamiltonian $\mathcal{H}(p, q)$. An important consequence of our proof is that, in Superstatistics, the identification of temperature with the kinetic energy is limited to the expectation of $\beta$ and cannot be used to measure the different temperatures in LTE or its fluctuations.

It is shown that a consistent application of Bayesian updating from a prior probability density to a posterior using evidence in the form of expectation constraints leads to exactly the same results as the application of the maximum entropy principle, namely a posterior belonging to the exponential family. The Bayesian updating procedure presented in this work is not expressed as a variational principle, and does not involve the concept of entropy. Therefore it conceptually constitutes a complete alternative to entropic methods of inference.